Zayed is helping his classmates get ready for their math test by making them identical packages of pencils and calculators. He has $72$ pencils and $24$ calculators and he must use all of the pencils and calculators. If Zayed creates the greatest number of identical packages possible, how many pencils will be in each package?
Solution: In order to know how many packages Zayed can make, we need a number that is a factor of ${72}$ and ${24}$, so that the ${72}$ pencils and the ${24}$ calculators can be divided up evenly. So, if there were $\gray{2}$ packages, there would be ${72} \div \gray{2} = 36$ pencils and ${24} \div \gray{2} = 12$ calculators in each package. This creates identical packages, but it isn't the greatest number of packages! To find the greatest number of identical packages, we want to find the greatest common factor of ${72}$ and ${24}$. To do so, let's find factors of ${72}$ and ${24}$. ${72}$ : $1,2, 3,4,6, 8, 9, 12, 18, {24}, 36, 72$ ${24}$ : $1,2, 3, 4, 6, 8, 12, {24}$ The greatest common factor of ${72}$ and ${24}$ is ${24}$. In math notation this looks like: $ \text{gcf}({72}, {24}) = {24}$. The greatest number of identical packages that Zayed can make is ${24}$. To find the number of pencils in each package, we need to divide the total number of pencils by the number of packages: $ {72} \div {24} = 3.$ There will be $3$ pencils in each package.